Mathematics Question and Solution
Notice, AB = 4 units is the height of the inscribed regular pentagon.
Calculating Area Blue.
a = ⅕(180(5-2))
a = 108°
a is the single interior angle of the inscribed regular pentagon.
b = ½(a)
b = ½(108)
b = 54°
c = ½(180-108)
c = ½(72)
c = 36°
d = b-c
d = 54-36
d = 18°
Therefore;
tan18 = e/4
e = 1.2996787849 units.
f = 2e
f = 2*1.2996787849
f = 2.5993575699 units.
f is the side length of the inscribed regular pentagon.
g = (4-r) units.
g is the radius of the inscribed circle.
r is the radius of the ascribed circle.
Calculating r.
r² = 1.2996787849²+(4-r)²
r² = 1.6891649439+16-8r+r²
8r = 17.6891649439
r = 2.211145618 units.
Again, r is the radius of the ascribed circle.
It implies;
g = (4-r)
And r = 2.211145618 units.
g = 4-2.211145618
g = 1.788854382 units.
Again, g is the radius of the inscribed circle.
h = 180-54-54
h = 72°
Area Blue is;
Area sector with radius 2.211145618 units and angle - Area triangle with height 2.211145618 units and base 2.211145618sin72 units + 2(area triangle with height 2.211145618 units and base 1.788854382sin36 units - Area sector with radius 1.788854382 units and angle 72°
= (72π*2.211145618²/360)-(0.5*2.211145618²sin72)+2(0.5.2.211145618*1.788854382sin36)-(72π*1.788854382²/360)
= 3.0719529341-2.3249360896+2.3249360896-2.0106192983
= 1.0613336358 square units.
≈ 1.06 square units to 2 decimal places.
